The motion of elongated gas bubbles in vertical pipes has been studied extensively over the past century. A number of empirical and numerical correlations have emerged out of this curiosity; amongst them, analytical solutions have been proposed. A review of the major results and resolution methods based on a potential flow theory approach is presented in this article. The governing equations of a single elongated gas bubble rising in a stagnant or moving liquid are given in the potential flow formalism. Two different resolution methods (the power series method and the total derivative method) are studied in detail. The results (velocity and shape) are investigated with respect to the surface tension effect. The use of a new multi-objective solver coupled with the total derivative method improves the research of solutions and demonstrates its validity for determining the bubble velocity. This review aims to highlight the power of analytical tools, resolution methods and their associated limitations behind often well-known and wide-spread results in the literature.

中文翻译：

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垂直管道中长孤立气泡速度和形状的势流解综述

在过去的一个世纪里，人们对垂直管道中细长气泡的运动进行了广泛的研究。出于这种好奇，出现了许多经验和数值相关性。其中，已经提出了分析解决方案。本文介绍了基于势流理论方法的主要结果和解决方法的回顾。在势流形式中给出了在静止或移动液体中上升的单个细长气泡的控制方程。详细研究了两种不同的分辨方法（幂级数法和全导数法）。就表面张力效应对结果（速度和形状）进行了研究。使用一种新的多目标求解器与全导数方法相结合，改进了对解的研究，并证明了其在确定气泡速度方面的有效性。这篇综述旨在强调分析工具的力量、分辨率方法及其在文献中通常众所周知和广泛传播的结果背后的相关局限性。